From English to Logic: Context-Free Computation of...

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From English to Logic: Context-Free Computation of "Conventional" Logical Translation 1 Lenhart K. Schubert Department of Computing Science University of Alberta Francis Jeffry Pelletier Department of Philosophy University of Alberta Edmonton, Canada T6G 2H1 We describe an approach to parsing and logical translation that was inspired by Gazdar's work on context-free grammar for English. Each grammar rule consists of a syntactic part that specifies an acceptable fragment of a parse tree, and a semantic part that specifies how the logical formulas corresponding to the constituents of the fragment are to be combined to yield the formula for the fragment. However, we have sought to reformulate Gazdar's semantic rules so as to obtain more or less 'conventional' logical translations of English sentences, avoiding the interpretation of NPs as property sets and the use of intensional functors other than certain propositional operators. The reformulat- ed semantic rules often turn out to be slightly simpler than Gazdar's. Moreover, by using a semantically ambiguous logical syntax for the preliminary translations, we can account for quantifier and coordinator scope ambiguities in syntactically unambiguous sentences without recourse to multiple semantic rules, and are able to separate the disambiguation process from the operation of the parser-translator. We have implemented simple recur- sive descent and left-corner parsers to demonstrate the practicality of our approach. 1. Introduction Our ultimate objective is the design of a natural language understanding system whose syntactic, se- mantic and pragmatic capabilities are encoded in an easily comprehensible and extensible form. In addi- tion, these encodings should be capable of supporting efficient algorithms for parsing and comprehension. In our view, the achievement of the former objec- tive calls for a careful structural separation of the sub- systems that specify possible constituent structure (syntax), possible mappings from constituent structure to underlying logical form (part of semantics), and possible mappings from logical form to deeper, unam- biguous representations as a function of discourse context and world knowledge (part of pragmatics and l Submitted August 1981; revised July 1982. inference). This sort of view is now widely held, as evidenced by a recent panel discussion on parsing issues (Robinson 1981). In the words of one of the panelists, "I take it to be uncontroversial that, other things being equal, a homogenized system is less preferable on both practical and scientific grounds than one that naturally decomposes. Practically, such a system is easier to build and maintain, since the parts can be designed, devel- oped, and understood to a certain extent in isolation... Scientifically, a decomposable sys- tem is much more likely to provide insight into the process of natural language comprehension, whether by machines or people." (Kaplan 1981) The panelists also emphasized that structural decom- position by no means precludes interleaving or paral- Copyright 1982 by the Association for Computational Linguistics. Permission to copy without fee all or part of this material is granted provided that the copies are not made for direct commercial advantage and the Journal reference and this copyright notice are included on the first page. To copy otherwise, or to republish, requires a fee and/or specific permission. 0362-613X/82/010026-19501.00 26 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982

Transcript of From English to Logic: Context-Free Computation of...

From English to Logic: Context-Free Computation of

"Conventional" Logical Translation 1

Lenhart K. Schubert

Department of Computing Science University of Alberta

Francis Je f f ry Pel let ier

Department of Philosophy University of Alberta

Edmonton, Canada T6G 2H1

W e describe an approach to parsing and logical translation that was inspired by Gazdar's work on context - free grammar for English. Each grammar rule consists o f a syntactic part that specifies an acceptable fragment of a parse tree, and a semantic part that specifies how the logical formulas corresponding to the constituents of the fragment are to be combined to yield the formula for the fragment. However, we have sought to reformulate Gazdar's semantic rules so as to obtain more or less 'conventional' logical translations of English sentences, avoiding the interpretation of NPs as property sets and the use of intensional functors other than certain propositional operators. The reformulat- ed semantic rules often turn out to be slightly simpler than Gazdar's. Moreover, by using a semantically ambiguous logical syntax for the preliminary translations, we can account for quantifier and coordinator scope ambiguities in syntactically unambiguous sentences without recourse to multiple semantic rules, and are able to separate the disambiguation process from the operation of the parser-translator. We have implemented simple recur- sive descent and left -corner parsers to demonstrate the practicality of our approach.

1. Introduction

Our ultimate objective is the design of a natural language understanding system whose syntactic, se- mantic and pragmatic capabilities are encoded in an easily comprehensible and extensible form. In addi- tion, these encodings should be capable of supporting efficient algorithms for parsing and comprehension.

In our view, the achievement of the former objec- tive calls for a careful structural separation of the sub- systems that specify possible constituent structure (syntax), possible mappings from constituent structure to underlying logical form (part of semantics), and possible mappings from logical form to deeper, unam- biguous representations as a function of discourse context and world knowledge (part of pragmatics and

l Submit ted Augus t 1981; revised July 1982.

inference). This sort of view is now widely held, as evidenced by a recent panel discussion on parsing issues (Robinson 1981). In the words of one of the panelists,

"I take it to be uncontroversial that, other things being equal, a homogenized system is less preferable on both practical and scientific grounds than one that naturally decomposes. Practically, such a system is easier to build and maintain, since the parts can be designed, devel- oped, and understood to a certain extent in isolation... Scientifically, a decomposable sys- tem is much more likely to provide insight into the process of natural language comprehension, whether by machines or people." (Kaplan 1981)

The panelists also emphasized that structural decom- position by no means precludes interleaving or paral-

Copyright 1982 by the Associa t ion for Computa t iona l Linguistics. Permiss ion to copy without fee all or part of this material is granted provided that the copies are not made for direct commercia l advantage and the Journal reference and this copyright notice are included on the first page. To copy otherwise, or to republish, requires a fee a n d / o r specific permission.

0362-613X/82/010026-19501.00

26 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982

Lenhart K. Schuber t and Francis Jeffry Pelletier From English to Logic

lelism of the processes that draw on the various kinds of linguistic and non-linguistic knowledge.

Note that we are making a distinction be tween the logical form that corresponds directly to surface struc- ture on the one hand, and an unambiguous deeper representat ion on the other. Indeed, at the level of logical fo rm our theory of logical t ranslat ion admits ambiguities in all of the formal building blocks (terms, functions, predicates, connectives, and quantifiers), as well as in the scopes of quantifiers and coordinators. For example, logical-form translat ions may contain terms such as Mary2 and < t h e l (little2 g i r l3)>, ambi- guous be tween various referents (e.g., MARY5 and MARY17) , and quasi-predicates such as has3, good2, cold5, and recovers l , ambiguous be tween various proper predicates (e.g., has3: OWNS1, A F F L I C T E D - W I T H . . . . ; good2: V I R T U O U S , G O O D - T A S T I N G . . . . ; cold5: C O L D 1 , E M O T I O N L E S S . . . . ; and r ecove r s l : R E - C O V E R S 1 , R E G A I N S . . . . ). In other words, we do not regard the logical form of a sentence as fully determining its meaning - not even its ' l i teral ' mean- ing; rather , its meaning is de te rmined by its logical form along with the context of its ut terance. Thus "She is becoming cold" might convey on one occasion that Lady Godiva is beginning to feel cold, on another that Queen Victoria is becoming emotionless, and on a third that Mount St. Helens is cooling off; but the logical form does no more than specify the feminine gender of the referent and its proper ty of "becoming cold (in some sense) at the time of u t terance" . Our pr imary concern in this paper will be with the semantic rules that define immediate logical form, although we a t tempt to define this form in a way that minimizes the remaining gap to the deeper representat ion.

All the experience gained within AI and linguistics suggests that bridging this final gap will be very diffi- cult. Some would take as their lesson that research efforts should concentra te on the last, pragmat ic phase of comprehension, where ' the real problems ' lie. We believe on the contrary that the only way to make the pragmat ic problems tractable is to have a precise con- ception of the consti tuent structure and logical form of the natural language input, in terms of which the prag- matic operat ions can in turn be precisely formulated.

In AI research, the object ives of clarity and exten- sibility have often been sacrificed to immediate per- formance goals. One reason for this may have been the need to establish the credibil i ty of a relat ively young and controversial discipline. In any case, the state of linguistic theory until fairly recently left no real al ternatives. The t ransformat iona l g rammars whose study dominated theoretical linguistics seemed a poor prospect even for the limited goal of describing natural language syntax, because of the subt le ty of t ransformat ional rules and supplementary devices such as co- indexing procedures , filters and constra ints on

movemen t , and the complexi ty of their interact ions. Moreover , the prospects for writing efficient t ransfor- mational parsers seemed poor, given that t ransforma- tional grammars can in principle generate all recursive- ly enumerable languages. But most importantly, gen- erative grammarians developed syntactic theories more or less independent ly of any semantic considerations, offering no guidance to AI researchers whose pr imary object ive was to compute 'mean ing represen ta t ions ' for natural language ut terances . Katz and Fodor ' s markerese (Katz & Fodor 1963) was patent ly inade- quate as a meaning representa t ion language f rom an AI point of view, and Genera t ive Semantics (Lakoff 1971) never did develop into a formal theory of the relation be tween surface form and meaning.

Theoret ical linguistics took an important new turn with the work of Montague on the logic of English and later expansions and variants of his theory (e.g., see Thomason 1974a, Partee 1976a, and Cresswell 1973). According to Montague g rammar the correspondence be tween syntactic structure and logical form is much simpler than had general ly been supposed: to each lexeme there corresponds a logical term or functor and to each rule of syntact ic composi t ion there corre- sponds a structurally analogous semantic rule of logical composit ion; this is the so-called rule-to-rule hypothe- sis [Bach 1976]. 2 Fur thermore , the translations of all consituents of a particular syntactic category are as- signed formal meanings of the same set- theoret ic type; for example , all NPs, be they names or definite or indefinite descriptions, are taken to denote p rope r ty sets. Crucially, the formal semant ics of the logical t ranslat ions produced by the semant ic rules of Mo- ntague grammar accords by and large with intuitions about entailment, synonymy, ambiguity and other se- mantic phenomena.

2 Interest ingly enough, this linguistic hypothes is was ant icipat- ed by K n u t h ' s work on the semant ics of at t r ibute g rammars (Knuth 1968). Schwind (1978) has applied Knu th ' s insights to the devel- opment of a formal basis for ques t ion answer ing sys tems, ant icipat- ing some of the work by Gazda r and o the rs on which our own efforts are founded.

There is also some similarity be tween the rule-to-rule hypothe- sis and the rule-based approach to the interpreta t ion of syntact ic s t ructures that emerged within AI during the 1960 's and early 70's . The idea of pairing semant ic rules with phrase s t ructure rules was at the hear t of D E A C O N (Craig et al. 1966), a sys tem based on F. B. T h o m p s o n ' s proposal to formalize English by limiting its subject ma t t e r to wel l -def ined compu te r m e m o r y s t ruc tu res ( T h o m p s o n 1966). However , D E A C O N ' s seman t i c rules pe r fo rmed direct semant ic evaluat ion of sorts (via computa t ions over a data base) rather than cons t ruc t ing logical t ranslat ions. The sys tems of Wino- grad (1972) and Woods (1977) cons t ruc ted input t ransla t ions prior to evaluat ion, using semant ic rules associated with particular syn- tactic s tructures. However , these rules nei ther cor responded one- to -one to syntac t ic rules nor l imited in terpre t ive opera t ions to composi t ion of logical express ions; for example, they incorporated tests for selectional restr ict ions and other forms of inference, with unrestr ic ted use of the computa t iona l power of LISP.

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Lenhar t K. Schubert and Francis Jeffry Pelletier From English to Logic

The chief limitation of Montague's grammar was that it treated only very small, syntactically (though not semantically) simple fragments of English, and efforts were soon under way to extend the fragments, in some cases by addition of a transformational com- ponent (Partee 1976b, Cooper & Parsons 1976). At the same time, however, linguists dissatisfied with transformational theory were beginning to develop non-transformational alternatives to traditional genera- tive grammars (e.g., Peters & Ritchie 1969, Bresnan 1978, Lapointe 1977, Brame 1978, Langendoen 1979). A particularly promising theory that emerged from this development, and explicitly incorporates Montague's approach to semantics, is the phrase struc- ture theory advanced by Gazdar and others (Gazdar 1980, 1981, Gazdar, Pullum & Sag 1980, Gazdar & Sag 1980, Sag 1980, Gazdar, Klein, Pullum & Sag, to appear). The theory covers a wide range of the syn- tactic phenomena that have exercised transformation- alists from Chomsky onward, including subcategoriza- tion, coordination, passivization, and unbounded de- pendencies such as those occurring in topicalization, relative clause constructions and comparatives. Yet the grammar itself makes no use of transformations; it consists entirely of phrase structure rules, with a node- admissibility rather than generative interpretation. For example, the rule [(S) (NP) (VP)] states that a frag- ment with root S, left branch NP and right branch VP is an admissible fragment of a syntactic tree. 3 Such phrase structure rules are easy to understand and per- mit the use of efficient context-free parsing methods. Moreover, the grammar realizes the rule-to-rule hy- pothesis, pairing each syntactic rule with a Montague- like semantic rule that supplies the intensional logic translation of the constituent admitted by the syntactic rule.

It has long been assumed by transformationalists that linguistic generalizations cannot be adequately captured in a grammar devoid of transformations. Gazdar refutes the assumption by using metagrammat- ical devices to achieve descriptive elegance. These devices include rule-schemata (e.g., coordination sche- mata that yield the rules of coordinate structure for all coordinators and all syntactic categories), and metarules (e.g., a passive metarule that takes any transitive-VP rule as 'input' and generates a corre- sponding passive-VP rule as 'output' by deleting the

3 We use traditional ca tegory symbols in our exposi t ion, occa- sionally followed by supp lementa ry features , e.g., (V T R A N ) for transi t ive verb. Gazdar actually a s sumes a two-bar X sys tem (e.g., see Bresnan 1976, Jackendof f 1977) that dis t inguishes be tween X, ,~ and X categories (e.g., ~ , V, and V, equivalent to the tradit ional S, VP and V respect ively) and employs complex symbo l s whose first componen t specifies the ' numbe r of bars ' and whose second componen t supplies a feature bundle encoding syntact ic category, subcategor izat ion, and morphosyn tac t i c and morphological in forma- tion.

object NP from the input rule and appending an op- tional by-PP). Although metarules resemble trans- formational rules, they map rules into rules rather than trees into trees, leaving the grammar itself context- free. Another key innovation is the use of categories with 'gaps', such as NP/PP, denoting a NP from which a PP has been deleted (not necessarily at the top lev- el). A simple metarule and a few rule schemata are used to introduce rules involving such derived categor- ies, elegantly capturing unbounded dependencies.

The character of the syntactic theory will become clearer in Section 4, where we supply a sampling of grammatical rules (with our variants of the semantic rules), along with the basic metarule for passives and the coordination schemata. First, however, we would like to motivate our attempt to reformulate Gazdar's semantic rules so as to yield 'conventional ' logical translations (Section 2), and to explain the syntactic and semantic idiosyncrasies of our target logic (Section 3).

By 'conventional' logics we mean first-order (and perhaps second-order) predicate logics, augmented with a lambda operator, necessity operator, proposi- tional attitude operators and perhaps other non- extensional propositional operators, and with a Kripke- style possible-worlds semantics (Hughes & Cresswell 1968). 4 The logic employed by Montague in his first formal fragment of English comes rather close to what we have in mind (Montague 1970a), while the inten- sional logics of the later fragments introduce the un- conventional features we hope to avoid (1970b,c). It is the treatment in these later fragments that is usually referred to by the term "Montague grammar". (For a detailed discussion of the distinction between conven- tional logics in the above sense and intensional logics, see Guenthner 1978).

We should stress that it is semantics, not syntax, which is the crux of the distinction. We shall take certain liberties with conventional logical syntax, align- ing it more nearly with the surface structure; but this will not lead to major departures from conventional semantics. For example, our syntax of terms allows syntactically unfamiliar formulas such as

[<alll man2> mortal3].

4 We admit predicate modif iers and some second-order predi- cate cons tan t s into our logical vocabulary, and may ul t imately want to employ a ful l-f ledged second-order logic, in view of such sen- t ences as " E v e r y good genera l has at least some of N a p o l e o n ' s qual i t ies" . On the o ther hand , we m a y pare down ra ther t h an expand the logical appara tus , opt ing for a logic that t reats proper- ties, proposi t ions and o ther in tensional ent i t ies as f i rs t -order indi- viduals. This type of t rea tment , which avoids the unwan ted identi ty of logically equivalent proposi t ions, appears to be gaining currency (e.g. , Fodor 1978, M c C a r t h y 1979, T h o m a s o n 1980, Chierch ia 1981). Some minor ad jus tmen t s would be required in our rules of logical t ranslat ion.

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But the formula derives its interpretat ion from its sti- pulated logical equivalence to

(alll x:[x man2])[x mortal3],

which may in turn become

Vx[[x HUMAN] = > [x MORTAL]] ,

after disambiguation. 5

2. Intensional and 'Conventional' Translations

We should emphasize at the outset that our objec- tive is not to impugn Montague grammar, but merely to make the point that the choice between intensional and conventional translations is as yet unclear. Given that the conventional approach appears to have cer- tain advantages, it is worth finding out where it leads; but we are not irrevocably committed to this approach. Fortunately, the translation component of a parser for a Gazdar-style grammar is easily replaced.

Montague grammarians assume that natural lan- guages closely resemble formal logical systems; more specifically, they postulate a strict homomorphism from the syntactic categories and rules of a natural language to the semantic categories and rules required for its formal interpretation. This postulate has led them to an analysis of the logical content of natural language sentences which differs in important respects from the sorts of analyses traditionally employed by philosophers of language (as well as linguists and AI researchers, when they have explicitly concerned themselves with logical content) .

The most obvious difference is that intensional logic translations of natural language sentences con- form closely with the surface structure of those sen- tences, except for some re-order ing of phrases, the introduction of brackets, variables and certain logical operators, and (perhaps) the reduction of idioms. For example, since the consti tuent structure of " John loves Mary" is

[John [loves Mary]],

the intensional logic translation likewise isolates a component translating the VP "loves Mary" , compos- ing this VP-translation with the translation of " J o h n " to give the sentence formula. By contrast, a conven- tional translation will have the structure

[John loves Mary],

in which " J o h n " and " M a r y " combine with the verb at the same level of consti tuent structure.

In itself, this difference is not important. It only becomes important when syntactic composition is as- sumed to correspond to function application in the semantic domain. This is done in Montague grammar

5 We consistently use infix form (with the predicate following its first argument) and square brackets for complete sentential formulas.

by resort to the Schoenf inkel -Church t rea tment of many-place funct ions as one-p lace funct ions (Schoenfinkel 1924, Church 1941). For example, the predicate " loves" in the above sentence is interpreted as a one-place function that yields a one-place function when applied to its argument (in this instance, when applied to the semantic value of "Mary" , it yields the function that is the semantic value of "loves Mary") . The resultant function in turn yields a sentence value when applied to its argument (in this instance, when applied to the semantic value of " John" , it yields the proposition expressed by " John loves Mary") . Thus, a dyadic predicator like " loves" is no longer interpreted as a set of pairs of individuals (at each possible world or index), but rather as a funct ion into functions. Similarly a triadic predicator like "gives" is interpreted as a function into functions into functions.

Moreover , the arguments of these functions are not individuals, because NPs in general and names in par- ticular are assumed to denote proper ty sets (or truth functions over properties) rather than individuals. It is easy to see how the postulate of syntact ic-semantic homomorphism leads to this further retreat from tradi- tional semantics. Consider Gazdar 's top-level rule of declarative sentence structure and meaning:

<10, [(S) (NP) (VP)], (VP' N P " ) > .

The first element of this triple supplies the rule num- ber (which we have set to 10 for consistency with the sample grammar of Section 4), the second the syntac- tic rule and the third the semantic rule. The semantic rule states that the intensional logic translation of the S-consti tuent is compounded of the VP-translation (as functor) and the NP-translat ion (as operand), where the latter is first to be prefixed with the intension op- erator A. In general, a primed syntactic symbol de- notes the logical translation of the corresponding con- stituent, and a double-primed symbol the logical trans- lation prefixed with the intension operator (thus, NP" stands for ANP') .

For example, if the NP dominates " J o h n " and the VP dominates "loves Mary" , then S' (the translation of S) is

(( loves ' AMary ' ) A John ' ) .

Similarly the translation of "Every boy loves Mary" comes out as

(( loves ' AMary ' ) A(every' b o y ' ) ) ,

given suitable rules of NP and VP formation. 6 Note the uniform treatment of NPs in the logical formulas, i.e., (every ' b o y ' ) is t reated as being of the same se- mantic category as John ' , namely the (unique) seman-

6 The exact function of the intension operator need not con- cern us here. Roughly speaking, it is used to bring meanings within the domain of discourse; e.g,, while an NP t denotes a property set at each index, the corresponding ANp~ denotes the entire NP intension (mapping from indices to property sets) at each index.

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Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

tic ca tegory corresponding to the syntact ic ca tegory NP. What is the set- theoret ic type of that category? Since (every v boy t) cannot be interpreted as denoting an individual (at least not without making the rules of semantic valuation for formulas depend on the struc- ture of the te rms they contain) , nei ther can John w. The solution is to regard NPs as denoting sets of prop- erties, where a proper ty determines a set of individuals at each index, and VPs as sets of such proper ty sets (or in functional terms, as truth functions over truth functions over propert ies) . Thus John v does not de- note an individual, but ra ther a set of proper t ies , namely those which John has; (every w boy t) denotes the set of propert ies shared by all boys, (a v boy w) the set of all propert ies possessed by at least one boy, and so on. It is not hard to see that the interpretat ion of VPs as sets of proper ty sets then leads to the appro- priate truth conditions for sentences. 7

With respect to our object ive of building a compre- hensible, expandable natural language unders tanding system, the simplicity of Gazda r ' s semantic rules and their one - to -one cor respondence to phrase s t ructure rules is extremely attractive; however , the semantics of the intensional logic translations, as sketched above, seems to us quite unnatural.

Admit tedly naturalness is par t ly a mat te r of famili- arity, and we are not about to fault Montague gram- mar for having novel fea tures (as some writers do, e.g., H a r m a n 1975). But Montague ' s semantics is at variance with pretheoret ical intuitions as well as philo- sophical tradition, as Montague himself acknowledged (1970c:268) . Intui t ively, names denote individuals (when they denote anything real), not sets of proper- ties of individuals; extensional transitive verbs express relat ions be tween pairs of individuals, not be tween pairs of p roper ty sets, and so on; and intuitively, quant if ied terms such as " e v e r y o n e " and " n o - o n e " simply don't bear the same sort of relationship to ob- jects in the world as names, even though the evidence for placing them in the same syntact ic ca tegory is overwhelming. Such object ions would carry no weight if the sole purpose of formal semantics were to pro- vide an explication of intuitions about truth and logical consequence, for in that area intensional logic is re- markably successful. But formal semantics should also do justice to our intuitions about the relationship be- tween word and object , where those intuit ions are clear - and intensional logic seems at odds with some of the clearest of those intuitions. 8

There is also a computa t ional object ion to inten- sional logic translations. As indicated in our introduc- tory remarks, a natural language unders tanding system must be able to make inferences that relate the natural language input to the sys tem's s tored knowledge and discourse model. A great deal of work in AI has fo- cused on inference during language unders tanding and on the organizat ion of the base of s tored knowledge on which the comprehens ion process draws. Almost all of this work has employed more or less convent ion- al logics for expressing the s tored knowledge. (Even such idiosyncratic formalisms as Schank 's conceptual depen~lency theory (Schank 1973) are much more akin to, say, first order modal logic than to any form of intensional logic - see Schubert 1976). H o w are in- tensional logic formulas to be connec ted up with stored knowledge of this convent ional type?

One possible answer is that the stored knowledge should not be of the convent iona l type at all, but should itself be expressed in intensional logic. H o w e v - er, the his tory of au tomat ic deduct ion suggests that higher-order logics are significantly harder to mecha- nize than lower -order logics. Deve lop ing eff icient inference rules and strategies for intensional logics, with their arbi t rar i ly complex types and their in ten- sion, extension and lambda abs t rac t ion opera to r s in addit ion to the usual modal opera tors , promises to be very difficult indeed.

Ano the r possible answer is that the intensional logic translations of input sentences should be post- processed to yield translat ions expressed in the lower- order, more convent ional logic of the sys tem's knowl- edge base. A difficulty with this answer is that dis- course inferences need to be computed 'on the fly ' to guide syntact ic choices. For example, in the sentences " John saw the bird without b inoculars" and " J o h n saw the bird without tail f ea the rs" the syntact ic roles of the preposi t ional phrases (i.e., whe ther they modi fy " s a w " or " the b i rd") can only be de termined by infer- ence. One could uncouple inference f rom parsing by comput ing all possible parses and choosing among the resultant translations, but this would be cumbersome and psychologically implausible at best.

As a final remark on the disadvantages of inten- sional t ranslat ions, we note that Mon tague g r ammar relies heavily on meaning postulates to deliver simple consequences, such as

A boy smiles - There is a boy;

7 This was the approach in Mon tague (1970b) and is adopted in Gazdar (1981a) . In another , less commonly adopted approach NPs are still in terpreted as sets of propert ies but VPs are in terpret - ed simply as propert ies, the t ruth condit ion for a sen tence being that the property denoted by the VP be in the set of propert ies deno ted by the NP (M on t ague 1970c, Cresswel l 1973) . In o ther words, the NP is thought of as predicat ing someth ing about the VP, rather than the o ther way around.

8 T h o m a s o n reminds us tha t " . . .we shou ld not forget the f i rmest and mos t irrefragable kind of da ta with which a semant ic theory mus t cope. The theory mus t h a r m o n i z e with the actual denota t ions taken by the express ions of natural languages , . . . " , but conf ines his fu r the r r emarks to s en tence deno ta t i ons , i.e., t ru th values (Thomason , 1974b:54) .

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Lenhar t K. Schubert and Francis Jeffry Pelletier From English to Logic

(in this instance an extensional izing postula te is re- quired for "smiles" - see Montague 1970c:263). A convent ional approach dispensing with postulates of this type would be preferable.

Having s tated our misgivings about Montague grammar, we need to confront the evidence in its fav- our. Are there compelling reasons for regarding sen- tential consti tuents as more or less directly and uni- formly in terpre table? In support of the aff i rmat ive, one can point out the simplicity and elegance of this s trategy f rom a logical point of view. More tellingly, one can cite its success record: it has made possible for the first time the formal characterizat ion of non- trivial f ragments of natural languages, with precisely def ined syntac t ic -semant ic mappings; and as one would hope, the formal semantics accounts for many cases of entailment, ambiguity, contradictoriness, and other semantic phenomena , including some of the sub- tlest arising f rom intensional locutions.

Concerning the simplicity of the strategy, we note that the connect ion between language and the world could be just as simple as Montague g rammar would have it, without being quite so direct. Suppose, for a moment , that people communicated in f i rs t-order logic. Then, to express that A1, Bill and Clyde were born and raised in New York, we would have to say, in effect, "Al was born in New York. A1 was raised in New York. Bill was born in New York . . . . Clyde was raised in New York ." The pressure to condense such redundant verbal izat ions would be great , and might well lead to ' over lay ' verbal izat ions in which lists enumerat ing the non- repea ted consti tuents were fit- ted into a common sentential matrix. In other words, it might lead to something like consti tuent coordina- tion. But unlike simple consti tuents, coordinated con- st i tuents would not be meaningful in isolation; they would realize their meaning only upon expansion of the embedding overlay verbalizat ion into a set of first- order formulas. Yet the connect ion be tween language and the world would remain simple, assuming that the syntactic relat ion be tween over lay verbal izat ions and their f irst-order translations were simple. It would be quite pointless to reeonstrue the semantics of the en- hanced language so as to align the denota t ions of names with the denotat ions of coordinated names, for example , as is done in Montague grammar . While formal ly simplifying the semant ic mapping function, such a move would lead to complex and counterintui- tive semantic types.

The success of Montague grammar in characterizing f ragments of natural languages, with a proper account of logical relations such as entailment, is indeed strong evidence in its favour. The only way of challenging this success is to offer an equally simple, equally via- ble alternative. In part , this paper is intended as a move in that direction. While we do not explicitly

discuss logical relat ions be tween the t ranslat ions of sentences, the kinds of translat ions produced by the sample g rammar in Section 4 should at least provide some basis for discussion. To the extent that the translations are of a conventional type (or easily con- ver ted to conventional form), the entai lment relations should be more or less self-evident.

There is one linguistic phenomenon , however , which deserves prel iminary comment since it might be thought to provide conclusive evidence in favour of Montague grammar, or at least in favour of the inten- sional t r ea tment of NPs. This concerns intensional verbs such as those in sentences (1) and (2), and per- haps (3):

(1) John looks for a unicorn, (2) John imagines a unicorn, (3) John worships a unicorn.

These sentences admit non-referen t ia l readings with respect to the NP "a unicorn" , i.e., readings that do not entail the existence of a unicorn which is the refer- ent of the NP. In intensional logic the nonreferent ia l reading of the first sentence would simply be

( ( looks-for t ^ ( a v unicornt ) ) ^ John t ) .

The formal semantic analysis of this formula turns out just as required; that is, its value can be " t r u e " or " f a l s e " (in a given possible world) i r respect ive of whether or not there are unicorns (in that world). The referent ia l reading is a little more complicated, but presents no difficulties.

It is the non-referent ia l reading which is t rouble- some for conventional logics. For the first sentence, there seems to be only one conventional translation, v iz . ,

3x[[John looks-for x] & [x unicorn]],

and of course, this is the referential reading. There is no direct way of represent ing the non-referent ia l read- ing, since the scope of a quant i f ier in convent ional logics is always a sentence, never a term.

The only possible escape f rom the difficulty lies in t ranslat ing intensional verbs as complex (non-a tomic ) logical expressions involving opaque sentent ial operators . 9 The extant l i terature on this subject sup- ports the view that a sat isfactory decomposi t ion can- not be supplied in all cases (Montague 1970c, Bennet t 1974, Partee 1974, Dowty 1978, 1979, Dowty, Wall & Peters 1981). A review of this literature would be out of place here; but we would like to indicate that the case against decomposi t ion (and hence against conven- tional translations) is not closed, by offering the fol-

9 With regard to our system-building objectives, such resort to lexical decomposition is no liability: the need for some use of lexi- cal decomposition to obtain "canonical" representations that facili- tate inference is widely acknowledged by AI researchers, and car- ried to extremes by some (e.g., Wilks 1974, Schank 1975).

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Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

lowing paraphrases of the three sample sentences. (Paraphrase (1)w is well-known, except perhaps for the particular form of adverbial (Quine 1960, Bennet t 1974, Par tee 1974), while (2 ) I - (3 ) '' are original). These could be formalized within a conventional logi- cal f ramework allowing for non-truth-funct ional sen- tential operators:

(1) ' John tries to find a unicorn (by looking around),

(2) I John forms a mental description which could apply to a unicorn,

(3) 1 John acts, thinks and feels as i f he worshipped a unicorn.

(3)" John worships an enti ty which he believes to be a unicorn.

In each case the operator that is the key to the trans- lation is italicized. Note that the original ambiguity of (1) and (2) has been preserved, but can now be con- strued as a quantifier scope ambiguity in the conven- tional fashion. In (3) 1 and (3)" the embedded "worsh ips" is to be taken in a veridical sense that entails the existence of the worshippee. It is important to unders tand that the translat ions corresponding to (3) 1 and (3)" would not be obtained directly by apply- ing the rules of the grammar to the original sentence; rather, they would be obtained by amending the direct translation, which is patently false for a hearer who interprets "worships" veridically and does not believe in unicorns. Thus we are presupposing a mechanism similar to that required to interpret metaphor on a Gricean account (Grice 1975). The notion of "acting, thinking and feeling as if. . ." may seem rather ad hoc, but appears to be applicable in a wide variety of cases where (arguably) non-intensional verbs of human ac- tion and attitude are used non-referential ly, as perhaps in " John is communing with a spirit", " John is afraid of the boogie-man in the att ic", or " John is tracking down a sasquatch". Formula t ion (3)" represents a more radical alternative, since it supplies an ac- ceptable interpretat ion of (3) only if the entity actually worshipped by John may be an ' imaginary unicorn ' . But we may need to add imaginary entities to our 'ontological stable' in any event, since entities may be explicitly described as imaginary (fictitious, hypothet i- cal, supposed) and yet be freely referred to in ordinary discourse. Also, sentences such as " John frequently dreams about a certain unicorn" (based on an example in Dowty, Wall and Peters 1981) seem to be untrans- latable into any logic without recourse to imaginary entities. Our paraphrases of (3) have the important advantage of entailing that John has a specific unicorn in mind, as intuitively required (in contrast with (1) and (2)). This is not the case for the intensional logic translation of (3) analogous to that of (1), a fact that led Bennet t to regard "worships" - correctly, we think - as extensional (Bennet t 1974).

In the light of these considerations, the convent ion- al approach to logical t ranslat ion seems well worth pursuing. The simplicity of the semantic rules to which we are led encourages us in this pursuit.

3. Syntactic and Semantic Preliminaries

The logical-form syntax provides for the format ion of simple terms such as

John1, x,

quantified terms such as

< s o m e l man2> , < t h e l (little2 b o y 3 ) > ,

simple predicate formulas such as

man2, loves3, P4,

compound predicate formulas such as

(loves2 Mary3), ((loves2 Mary3) John l ) , [Johnl loves2 Mary3],

modified predicate formulas such as

(bright3 red4), (passionately2 (loves3 Mary4)) ,

and lambda abstracts such as

~x[x shaves2 x], Xy[y expects2 [y wins4]].

Note the use of sharp angle brackets for quantif ied terms, square brackets or blunt angle brackets for compound predicate formulas, and round brackets for modified predicate formulas. (We explain the use of square brackets and blunt angle brackets below.) We also permit sentences (i.e., compound predicate formu- las with all arguments in place) as operands of senten- tial operators, as in

[[John5 loves6 Mary7] possible3], [Suel believes2 [John5 loves5 Mary6]], [ [Johnl feverish3] because4

[Johnl has5 malaria6]].

For coordinat ion of expressions of all types (quantifiers, terms, predicate formulas, modifiers, and sentential operators) we use sharp angle brackets and prefix form, as in

<or2 many l f ew3>, <and2 Jo h n l Bill3>,

<and4 (hugs2 Mary3) (kisses5 Sue6)>.

The resemblance of coordinated expressions to quanti- fied terms is intentional: in both cases the sharp angle brackets signal the presence of an unscoped operator (viz., the first element in brackets) to be scoped later on.

Finally, we may want to admit second-order predi- cates with f irst-order predicate arguments, as in

[Fidol little-for3 dog5], [bluel colour4],

[Xx[x kissesl Mary2] is-fun3],

though it remains to be seen whether such second- order predications adequately capture the meaning of English sentences involving implicit comparatives and nominalization.

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Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

Fuller explanations of several of the above features follow. In outline, we first delve a little further into the syntax and semantics of predicate formulas; then we discuss the sources and significance of ambiguities in the formulas.

Atomic sentences are of the form

[t n P t 1 ... tn_l], (equivalently, (P t 1 ... tn)),

where t I . . . . , t n are terms and P is a predicate con- stant, and the square brackets and blunt angle brack- ets distinguish infix and prefix syntax respectively. We regard this sentential form as equivalent to

[t n (...((e t 1) t 2) ... tn_ 1 )1, i.e., as obta ined by applying an n-ary predicate successively to n terms. For example,

[John loves Mary] = Cloves Mary John) = [ John Cloves Mary)] = ((loves Mary) John). 10

As in Montague grammar, this predicate application syntax helps to keep the rules of translation simple: in most cases the translation of a phrase is just the com- posit ion of the t ranslat ions of its top- level const i tu- ents. However , we saw earlier that a functional inter- pretat ion of predicate application leads to the interpre- ta t ion of predicates as te lescoped funct ion-va lued functions, whereas we wish to interpret predicates as n-ary relations (in each possible world) in the conven- tional way.

We can satisfy this requirement by interpreting predicate application not as function application, but ra ther as leftmost section of the associated relation at the value of the given argument. For example, let V denote the semantic valuation function (with a part icu- lar interpretat ion and possible world understood) and let

V(P) = {<a ,b , c> , < a , b , d > , <e , f ,g>} ,

V(x) = a, V(y) = b, and V(z) = d,

where P is a triadic predicate symbol, x, y, and z are individual constants or variables, and a, b . . . . . g are elements of the individual domain D. Then

V((P x)) = { < b , c > , < b , d > } ,

V((P x y)) = V(((P x) y) = { < c > , < d > } , and

V([z V x y]) = V((((P x) y) z)) = {<>} .

We use the convent ion { < > } = true, {} = false.

Lambda abstract ion can be defined compat ibly by

Vl(~,x~b) = {{d} X V i ( x : d ) (~b) I d • D},

where I is an interpretat ion, I(x:d) is an interpretat ion identical to I except that x denotes d, and X denotes Cartesian product (and a particular possible world is

10 We provide the double syntax for purely cosmetic reasons. In our use of the notat ion, express ions delimited by square brackets will generally be complete open or closed sentences , while expres- sions del imited by blunt angle brackets will be ' incomple te sen tences ' , i.e., predicates with one or more a rgument s miss ing (and denot ing a relation with adicity = number of miss ing a rguments ) .

understood). It can be verified that the usual lambda- conversion identities hold, i.e.,

()~x(P...x...) t) = (P...t...), and

P = )tx(P x) = ~,xhy(P x y) . . . . .

where P is a predicate of any adicity (including null, if we u s e { < > } X A = A for any set A).

As far as modif ied predicate formulas such as (bright3 red4) are concerned, we can interpret the modif iers as funct ions f rom n-ary relat ions to n-ary relations (perhaps with n restricted to 1).

We now turn to a considera t ion of the potent ia l sources of ambiguity in the formulas. One source of ambiguity noted in the Int roduct ion lies in the primi- tive logical symbols themselves, which may correspond ambiguously to various proper logical symbols. The ambiguous symbols are obta ined by the translator via the first stage of a two-s tage lexicon (and with the aid of morphological analysis, not discussed here). This first stage merely distinguishes the formal logical roles of a lexeme, supplying a distinct (but in general still ambiguous) symbol or compound expression for each role, along with syntactic information. For example, the entry for " r e c o v e r " might distinguish (i) a predi- cate role with prel iminary translation " r ecov e r s - f ro m" and the syntactic information that this is a V admissi- ble in the rule that expands a VP as a V optionally fol lowed by a (PP from); (this information is supplied via the appropr ia te rule number) ; and (ii) a predicate role with prel iminary t ranslat ion " r e c o v e r s " and the syntactic informat ion t ha t this is a V admissible in the rule that expands a VP as a V followed by an NP.

Having obta ined a prel iminary translation of a lex- eme in keeping with its apparen t syntact ic role, the translator affixes an index to it which has not yet been used in the current sentence (or if the translation is a compound expression, it affixes the same index to all of its primitive symbols). In this way indexed pre- l iminary t ranslat ions such as M a r y l , good2, and recovers3 are obtained. For example, the verb trans- lation selected for " r ecove r s " in the sentence context " J o h n recovers the so fa" would be recovers2, recovers - f rom2 being ruled out by the presence of the NP complement . The second stage of the lexicon sup- plies al ternat ive final t ranslat ions of the f i rs t -s tage symbols , which in the case of " r e c o v e r s " might be R E - C O V E R S , R E G A I N S , and so on. Naturally, the processors that choose among these final symbols would have to draw on knowledge stored in the propo- sitional data base and in the represen ta t ion of the discourse context.

A second source of ambigui ty lies in quant i f ied terms. The sentence

Someone loves every man

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Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

illustrates a quantifier scope ambiguity arising from a syntactically unambiguous construction. Its logical- form translation is

[<somel one2> loves3 <every4 man5>],

wherein the relative scopes of the quantifiers somel and every4 are ambiguous. Quantified terms are in- tended to be 'extracted' in the postprocessing phase to positions left-adjacent to sentential formulas (which may already be prefixed with other quantifiers). A new variable is introduced into each extracted quanti- fier expression, the angle brackets are changed to round brackets, and the new variable is substituted for all occurrences of the extracted term. (Thus the level of extraction must be 'high' enough to encompass all of these occurrences.) In the above formula, quantifier extraction reveals the implicit ambiguity, yielding ei- ther

(somel x:[x one2])(every4 y:[y man5])[x loves3 y] or

(every4 y:[y man5])(somel x:[x one2])[x loves3 y],

depending on the order of extraction.

Assuming that somel and every4 correspond to the standard existential and universal quantifiers, these translations could be further processed to yield

~x[[x one2] & Vy[[y man5] = > [x loves3 y]]] and Vy[[y man5] = > 3x[[x one2] & [x loves3 y]]].

However, we may not implement this last conversion step, since it cannot be carried out for all quantifiers. For example, as Cresswell remarks, "most A's are B's" cannot be rendered as "for most x, either x is not an A or x is a B" (Cresswell 1973: 137). (Consider, for instance, A = dog and B = beagle; then the last state- ment is true merely because most things are not dogs - irrespective of whether or not most dogs are in fact beagles.) It appears from recent work by Goebel (to appear) that standard mechanical inference methods can readily be extended to deal with formulas with restricted quantifiers.

A third source of ambiguity lies in coordinated expressions. For example, the logical form of the sentence "Every man loves Peggy or Sue" is

[<everyl man2> loves3 <or5 Peggy4 Sue6>],

which is open to the readings

(everyl x:[x man2])[ [x loves3 Peggy4] or5 [x loves3 Sue6]]

and [(everyl x:[x man2])[x loves3 Peggy4]

or5 (everyl x:[x man2])[x loves3 Sue6]].

The postprocessing steps required to scope coordina- tors are similar to those for quantifiers and are illus- trated in Section 4.1 l

An important constraint on the disambiguation of the basic symbols as well as quantified terms and coor- dinated expressions is that identical expressions (i.e.,

expressions with identical constituent structure, includ- ing indices) must be identically disambiguated. For example, "John shaves himself" and "John shaves John" translate respectively into

[Johnl hx[x shaves2 x]] = [Johnl shaves2 Johnl] , and

[Johnl shaves2 John3].

The stated constraint ensures that both occurrences of Johnl in the first formula will ultimately be replaced by the same unambiguous constant. Similarly "Someone shaves himself" and "Someone shaves someone" translate initially into

[<somel one2> shaves3 <somel one2>] and

[<somel one2> shaves3 <some4 one5>]

respectively, and these translations become

(somel x:[x one2])[x shaves3 x] and

(somel x:[x one2])(some4 y:[y one5])[x shaves3 y]

respectively after quantifier extraction. Note that the two occurrences of <somel one2> in the first formula are extracted in unison and replaced by a common variable. Indexing will be seen to play a similar role in the distribution of coordinators that coordinate non- sentential constituents.

By allowing the above types of ambiguities in the logical form translations, we are able to separate the problem of disambiguation from the problems of pars- ing and translation. This is an important advantage, since disambiguation depends upon pragmatic factors. For example, "John admires John" may refer to two distinct individuals or just to one (perhaps whimsical- ly), depending on such factors as whether more than one individual named John has been mentioned in the current context. Examples involving ambiguities in nouns, verbs, determiners, etc., are easily supplied. Similarly, the determination of relative quantifier scopes involves pragmatic considerations in addition to level of syntactic embedding and surface order. This is true both for explicit quantifier scope ambiguities such as in the sentence "Someone loves every man", and for scope ambiguities introduced by decomposi- tion, such as the decomposition of "seeks" into

hyhx[x tries [x finds y]],

as a result of which a sentence like

John seeks a unicorn

admits the alternative translations

3x[[x unicorn] & [John tries [John finds x]]], and [John tries 3x[[x unicorn] & [John finds x]]],

neglecting indices. It is simpler to produce a single output which can then be subjected to pragmatic post-

11 If f i rs t-order predicates are to be allowed as arguments of second-order predicates, then quantifier and coordinator scoping of the following types must also be allowed: [P . . .<Q R>. . . ] -~ ~kx(Q y:[y Rl)[x P...y...], < C P R > ~ ~kx[[x PI C [x R]].

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Lenhar t K. Schuber t and Francis Jeffry Pelletier From English to Logic

processing to determine likely quantifier scopes, than to generate all possible orderings and then to make a pragmatic choice among them. Much the same can be said about scoping of coordinators.

We also note that a g rammar designed to generate all possible unambiguous translations of English phras- es and sentences would have to supply multiple seman- tic rules for certain syntactic rules. For example, no one semantic rule can translate a quant i f ier-noun com- bination (rule 3 in Section 4) so as to deliver both readings of "Someone loves every m a n " upon combi- nation of the verb translation with the translations of the NPs. Our use of an ambiguous logical form pre- serves the rule-to-rule hypothesis.

4. Sample Grammar

Our syntactic rules do not depart significantly f rom Gazdar ' s . The semant ic rules formal ly resemble Gazdar ' s as well, but of course produce conventional ly interpretable translations of the type described in the preceding section. As in Gazda r ' s semant ic rules, consti tuent translations are denoted by pr imed catego- ry symbols such as N P ' and V ' . The semantic rules show how to assemble such translat ions (along with the occasional variable and lambda opera tor) to form the translations of larger constituents. The transla- tions of individual lexemes are obta ined as described above.

In operat ion, the t ranslator generates the minimum number of brackets consis tent with the nota t ional equivalences stated earlier. For example, in assem- bling [NP ' V P ' ] , with N P ' = J o h n l and VP t = [loves2 Mary3] , the result is

[ Johnl loves2 Mary3],

ra ther than

[Johnl (loves2 Mary3)].

Also, in binding a variable with lambda, the translator replaces all occurrences of the variable with a previ- ously unused variable, thus minimizing the need for later renaming. Finally, it per forms lambda conver- sions on the fly. For example, the result of assembling [NP ' V P ' ] with N P ' = J o h n l and

VPV= ?~x[x shaves2 x], is

[ Johnl shaves2 John l ] .

The rules that follow have been adapted f rom Gaz- dar (1981a). Note that each rule that involves a lexi- cal category such as PN, N or V is accompanied by a specification of the subset of lexical items of that cate- gory admissible in the rule. This feature is particularly impor tan t for verb subcategor iza t ion . In addition, each rule is followed by (a) a sample phrase accepted by the rule, (b) an indication of how the logical t rans- lation of the phrase is obtained, and possibly (c) some words of further explanation.

<I, [(NP) (PN)], PN'>, PN(1) = [John, Mary, New York .... ]

(a) Mary

(b) with PN' : Mary6, NP' becomes Mary6.

<2, [(AN) (ADJP) (N)], (ADJP' N')>, N(2) = [boy, game, noise,

(a) little boy

(b) with ADJP' : little2, N' = boy3,

AN' becomes (little2 boy3);

(c) "little" is taken as a predicate modifier. ]2

<3, [(NP) (Q) (AN)], <Q' AN'>>, Q(3) : [a, the, all, many,

(a) the little boy

(b) with Q' = thel, AN' = (little2 boy3),

NP' -> <thel (little2 boy3)>.

<4, [(PP to) (to) (NP)], NP'>

(a) to Mary

(b) with NP' = Mary6, PP' -> Mary6;

(c) PP verb complements have the same meaning as their NP,

<5, [(VP) (V)], V'>, V(5) = {run, smile, disappear .... ]

(a) smiles

(b) with V' = smiles4, VP' -> smiles4.

• . .]

...]

as per Gazdar (1981a).

12 Siegel (1979) argues rather persuasively that measure adjec- tives, unlike genuine predicate modif iers such as " c o n s u m m a t e " , actually combine with terms. For such adjectives we might employ the semant ic rule ~.x[[x A D J P ' ] & [x N ' ] ] ; in the case of "l i t t le",

we would use A D J P ' = (little-for P), where P is an indeterminate predicate to be replaced pragmatical ly by a compar ison-c lass predi- cate. Thus the t ranslat ion of "little boy" (neglect ing indices) would be ~kx[[x little-for PI & [x boyl].

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Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

<6, [(VP) (V) (NP) (PP to)], (V' PP' NP')>

V(6) = {give, hand, tell .... ]

(a) gives Fido to Mary

(b) with V' = gives4, NP' : Fido5, PP' : Mary6, VP' -> (gives4 Mary6 Fido5>.

<7, [(VP INF) (to) (VP BASE)], VP'>

(a) to give Fido to Mary

(b) with VP' : (gives4 Mary6 Fido5>, the resultant infinitive has the

same meaning.

<8, [(VP) (V) (VP INF)], Ax[x V' [x VP']]>, V(8) : {want, expect, try .... ]

(a) wants to give Fido to Mary

(b) with V' : wants2, VP' = {gives4 Mary6 Fido5],

VP' -> Ax3[x3 wants2 [x3 gives4 Mary6 Fido5]];

(c) The formal lambda variable x given in the semantic rule has been replaced by

the new variable x3. Two pairs of square brackets have been deleted, in

accordance with the simplification rules stated earlier.

<9, [(VP) (V) (NP) (VP INF)], (V' [NP' VP'])>, V(9) = {want, expect, imagine .... ]

(a) wants Bill to give Fido to Mary

(b) with V' : wants2, NP' : Bill3, VP' = (gives~ Mary6 Fido5),

VP' -> (wants2 {Bill3 gives4 Mary6 Fido5]).

<10, [(S DECL) (NP) (VP)], [NP' VP']>

(a) the little boy smiles

(b) with NP' = <the] (little2 boy3)> and VP' = smiles4, the result is

S' -> [<thel (little2 boy3)> smiles4]. After pragmatic postprocessing

to extract quantifiers, the result might be

S' = (thel x5:[x5 (little2 boy3)]) [x5 smiles4].

Further postprocessing to determine referents and disambiguate operators

and predicates might then yield

S' = [INDIVI7 SMILESl],

where INDIV17 is a (possibly new) logical constant unambiguously denoting

the referent of (the] x5:[x5 (little2 boy3)]) and SMILESl is an unambiguous

logical predicate. 13 If constant INDIV17 is new, i.e., if the context provided

no referent for the definite description, a supplementary assertion like

[INDIV17 (LITTLE2 BOYI)]

would be added to the context representation.

(a)' John wants to give Fido to Mary

(b)' with NP' = Johnl,

VP' : lx3[x3 wants2 [x3 gives4 Mary6 FidoS]],

S' -> [Johnl wants2 [Johnl gives4 Mary6 FidoS]];

(c) ' Note that Johnl becomes the subject of both the main clause and the

embedded (subordinate) clause.

The reader will observe that we have more or less fully traced the derivation and translation of the sen- tences "The little boy smiles" and "John wants to give Fido to Mary" in the course of the above examples.

The resultant phrase structure trees, with rule numbers and translations indicated at each node, are shown in Figs. 1 and 2.

13 Definite singular terms often serve as descriptions to be used

for referent determination, and in such cases it is the name of the

referent, rather than the descript ion itself , which is u l t imate ly wanted in the formula.

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Lenhar t K. Schuber t and Francis Je f f ry Pel le t ier From English to Logic

r u l e 1: NP' =PN'=dohnl

PN'=dohnl

I dohn

© r u l e 10: S' = [NP' VP' ]

= [dohnl wants2 [dohnl g i v e s 4 Mary6 F ido5 ] ]

r u l e 8: VP' = Xx[x V' [x V P ' ] I = ~x3[x3 wants2 [x3 g i ves4 Mary6 F i d o 5 ] ]

\ (vp IN© U

V' =wants2

I w a n t s

r u l e 7" (VP INF) '= (VP BASE)' = <g ives4 Mary6 Fido5>

/ to (_VP BASE)

r u l e 6" VP' = ( V' PP' NP' ) / ~ ( g i v e s 4 Mary6 Fido5>

( V BASE)

V' -g i ves4

I g i v e

r u l e 1" NP' =PN' = Fido5

PN' =Fido5

Fido

Cpp to) r u l e 4 PP' =NP'

= Mar't6

/ r u l e 1: NP'=PN'

= Mary6

I Mary

Figure 1. Phrase structure and translation of the sentence

"John wants to give Fido to Mary".

Amer ican Journal of Computa t iona l Linguist ics, Vo lume 8, Number 1, January-March 1982 37

Lenhart K. Schubert and Francis Jef f ry Pel let ier From English to Logic

r u l e = <the1

Q ' = t h e l

I t h e

Q r u l e 10: S' = [NP' VP' ]

= [ < t h e 1 ( l i t t l e 2 b o y 3 ) > s m i l e s 4 ]

3: NP'= <O' AN' > r u l e 5: VP' = ( l i t t l e 2 b o y 3 ) > = s m i l e s 4

s r n i l e s

r u l e 2: AN'= (ADJP' N' ) = ( l i t t l e 2 boy3)

r u l e n: ADdP' = ADd' N' =boy3 = l i t t l e 2 I

boy

ADd'= i t t l e 2

V,

l i t t l e

Figure 2. Phrase s t ructure and t rans la t ion of the sen tence

"The little boy smiles."

38 American Journal of Computat ional Linguist ics, Volume 8, Number 1, January-March 1982

Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

All of the above rules, as well as our versions of the remaining rules in Gazdar (1981a), are as simple as the intensional logic versions or simpler. For exam- ple, our semantic rule 8, i.e., Ax[x V' [x VP'] ] , may be contrasted with the corresponding rule suggested by Gazdar:

XP{P Xx[(V' A(Vp' XP(P x))) XP(P x)]}.

Here the lambda variable x, as in our formula, is used to feed a common logical subject to V ' (the transla- tion of the main verb) and to VP' (the translation of the embedded infinitive); the variables P and P, on the other hand, serve to ensure that the arguments of the V' and VP ' functions will be of the correct type. Our 'conventional ' rule is simpler because it makes no such use of lambda abstraction for type-raising and dispens- es with the intension operator.

Gazdar ' s approach to unbounded dependencies carries over virtually unchanged and can be illustrated with the sentence

To Mary John wants to give Fido.

Here the PP " to Mary" has been topicalized by ex- tract ion from " John wants to give Fido to Mary" , leaving a PP 'gap' at the extraction site. This 'gap' is syntactically embedded within the infinitive VP " to give Fido" , within the main VP "wants to give Fido" , and at the highest level, within the sentence " Jo h n wants to give Fido". In general, the analysis of un-

bounded dependencies requires derived rules for propa- gating 'gaps' from level to level and linking rules for creating and filling them. The linking rules are ob- tained from the correspondingly numbered basic rules by means of the metarule

[AXCY] ==> [A/BXC/BY],

where A, B and C may be any basic (i.e., non-slash) syntactic categories such that C can dominate B, and X, Y may be any sequences (possibly empty) of bas- ic categories. The linking rules for topicalization are obtained from the rule schemata

<I I, [B/B t] , h>, and

<12, [(S) B (S)/B], <AhS' B')>,

where B ranges over all basic phrasal categories, and t is a dummy element (trace). The first of these sche- mata introduces the free variable h as the translation of the gap, while the second lambda-abstracts on h and then supplies B' as the value of the lambda varia- ble, thus 'filling the gap' at the sentence level. At syntactic nodes intermediate between those admitted by schemata 11 and 12, the B-gap is transmitted by derived rules and h is still free.

Of the following rules, 6, 8, and 10 are the particu- lar derived rules required to propagate the PP-gap in our example and 11 and 12 the particular linking rules that create and fill it:

<11, [(PP to)/(PP to)

(a) t

(b) PP' -> h

t], h>

<6, [(VP)/(PP to) (V) (NP) (PP to)/(PP to)], (V' PP' NP')>

(a) give Fido

(b) with V' : gives5, NP' : Fido6, PP' = h,

VP' -> (gives5 h Fido6)

(c) Note that the semantic rule is unchanged.

<8, [(VP)/(PP to) (V) (VP INF)/(PP to)], Ax[x V'

(a) wants to give Fido

(b) with V' = wants3, VP' = (gives5 h Fido6),

VP' -> Ax4[x4 wants3 [x4 gives5 h Fido6]]

[x VP']]>

<10, [(S)/(PP to) (NP) (VP)/(PP to)], [NP' VP']>

(a) John wants to give Fido

(b) with NP' : John2,

VP' = Ax4[x4 wants3 Ix4 gives5 h Fido6]],

S' -> [John2 wants3 [John2 gives5 h Fido6]]

<12, (a) (b)

(c)

[(S) (PP to) (S)/(PP to)], (AhS' PP')>

To Mary John wants to give Fido

With S' as in 10 (b) above and PP' : Maryl,

S' -> [John2 wants3 [John2 gives5 Maryl Fido6] ] .

This translation is logically indistinguishable from the

translation of the untopicalized sentence. However, the

fronting of "to Mary" has left a pragmatic trace: the

American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 39

Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

corresponding argument Maryl has the lowest index, lower

than that of the subject translation John2 (assuming that

symbols are indexed in the order of occurrence of the

lexical items they translate). In subsequent pragmatic

processing, this feature could be used to detect the special

salience of Maryl, without re-examination of the superficial

sentence form.

Another example of a sentence that can be ana- lyzed by such methods, using relative clause rules simi- lar to those for topicalization, is

Every dog Mary wants to buy is small.

The rules analyze "Mary wants to buy" as an S /NP with translation

[Mary wants [Mary buys h]],

neglecting indices. A further rule reduces the S /NP to an R (relative clause), and its semantic part abstracts on h to yield the predicate

R' = Xh[Mary wants [Mary buys h]]

as the translation of the relative clause. The rules for NPs can be formulated in such a way that "every dog" will be translated as

<every kx[[x dog] & [x R]]>

where R is a free predicate variable that is replaced by the translation of the relative clause when the NP-R rule

<13, [(NP) (NP) (R)], <XRNP' R ' > >

is applied (cf., Gazdar 1981b; we have ignored multi- ple relative clauses). The resulting NP translation is

<every hx[[x dog] & [Mary wants [Mary buys x]]]>.

The translation of the complete sentence, after extrac- tion of the quantifier and conversion of the constraint on the universally quantified variable to an implicative antecedent, would be

¥y[[[y dog] & [Mary wants [Mary buys y]]]

= > [y (small P)]],

where P is an undetermined predicate (= dog, in the absence of contrary contextual information).

As a further illustration of Gazdar's approach and how easily it is adapted to our purposes, we consider his metarule for passives:

<[(VP)(V TRAN) (NP) X], (St N P " ) > = = >

<[(VP PASS) (V) X {(PP by)}], ~,p((~r p) pp")>;

i.e., "for every active VP rule that expands VP as a transitive verb followed by NP, there is to be a passive VP rule that expands VP as V followed by what, if anything, followed the NP in the active VP rule, fol- lowed optionally by a by-PP" (Gazdar 1981a). In the original and resultant semantic rules, (~" ...) represents the original rule matrix in which NP" is embedded; thus (~r p) is the result of substituting the lambda

variable P (which varies over NP intensions) for NP" in the original rule. Intuitively, the lambda variable 'reserves' the NP" argument position for later binding by the subject of the passive sentence. It can be seen that the metarule will generate a passive VP rule cor- responding to our rule 6 which will account for sen- tences such as "Fido was given to Mary by John". Moreover, if we introduce a ditransitive rule

<14, [ (VP) (V TRAN) (NP) (NP) ] , (V' NP' NP')>I4

to allow for sentences such as "John gave Mary Fido", the metarule will generate a passive VP rule that ac- counts for "Mary was given Fido by John", in which the indirect rather than direct object has been turned into the sentence subject.

The only change needed for our purposes is the replacement of the property variable P introduced by the metarule by an individual variable x:

...(~r NP' ) . . . . . > ...hx((~ r x) PP' )...

Once the subject NP of the sentence is supplied via rule 10, x is replaced by the translation of that NP upon lambda conversion.

Finally in this section, we shall briefly consider coordination. Gazdar has supplied general coordina- tion rule schemata along with a cross-categorical se- mantics that assigns appropriate formal meanings to coordinate structures of any category (Gazdar 1980b). Like Gazdar's rules, our rules generate logical-form translations of coordinated constituents such as

<and John Bill>, <or many few>, <and (hugs Mary) (kisses Sue)>,

echoing the surface forms. However, it should be clear from our discussion in Section 2 that direct inter- pretation of expressions translating, say, coordinated NPs or VPs is not compatible with our conventional conception of formal semantics. For example, no for- mal semantic value is assigned directly to the coordi- nated term in the formula

[<and John Bill> loves Mary].

Rather, interpretation is deferred until the pragmatic processor has extracted the coordinator from the em- bedding sentence (much as in the case of quantified

14 In the computational version of the semantic rules, primed symbols are actually represented as numbers giving the positions of the corresponding constituents, e.g., (1 2 3) in rule 14. Thus no ambiguity can arise.

40 Amer ican Journal of Compu ta t i ona l Linguist ics, Vo lume 8, Number 1, January -March 1982

Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

terms) and distributed the coordinated terms over duplicate copies of that sentence, yielding

[[John loves Mary] and [Bill loves Mary]].

We adopt the following coordination schemata without change. The superscript denotes sequences of

length > 1 of the superscripted element. The schema- ta are accompanied by examples of phrases they admit, along with (unindexed) translations. The bracketing in (a) and (a)' indicates syntactic structure.

<15, [(A ~) (~) (A)], A'>,

where A is any syntactic category and ~ E {and, or]

(a) and admires

(b) admires

(a)' or Mary

(b)' Mary

<16, [(A) (A)+ (A ~)], <~' A'A'...A'>>

(a) loves [and admires]

(b) <and loves admires>

(a)' [Fido Kim] [or Mary]

(b)' <or Fido Kim Mary>

<17, [(A) (A) (A ~)+], <~' A'A'...A'>>

(a) Fido [[or Kim] [or Mary]]

(b) <or Fido Kim Mary>

The order in which coordinators are extracted and distributed is a matter of pragmatic choice. However, a crucial constraint is that multiple occurrences of a particular coordinated expression (with particular ind- ices) must be extracted and distributed in a single

operation, at the level of a sentential formula whose scope encompasses all of those occurrences (much as in the case of quantifier extraction). The following examples illustrate this process.

(a) John loves and admires Fido or Kim

(b) [Johnl <and3 loves2 admires4> <or6 Fido5 Kim7>] ->

[[Johnl loves2 <or6 Fido5 Kim7>] and3

[Johnl admires4 <or6 Fido5 Kim7>]] ->

[[[Johnl loves2 Fido5] and3

[Johnl admires4 Fido5]] or6

[[Johnl loves2 Kim7] and3

[Johnl admires4 Kim7]]].

(c) Note that once the and3-conjunction has been chosen for initial

extraction and distribution, the simultaneous extraction and

distribution of both occurrences of the or6-disjunction at the

highest sentential level is compulsory. The resultant formula

expresses the sense of "John loves and admires Fido or loves

and admires Kim". Initial extraction of the or6-disjunction

would have led to the (implausible) reading "John loves Fido or

Kim and admires Fido or Kim" (which is true even if John loves

only Fido and admires only Kim).

(a) ' All men want to marry Peggy or Sue

(b)' [<alll man2> wants3 [<alll man2> marries4 <or6 Peggy5 Sue7>]]->

(alll x: [x man2]) Ix wants3 [x marries4 <or6 Peggy5 Sue7>]] ->

(alll x: [x man2]) [x wants3

[[x marries4 Peggy5] or6 [x marries4 Sue7]]].

(c) ' In the second step above, the coordinator or6 might instead

have been raised to the second highest sentential level, yielding

(alll x: [x man2]) [Ix wants3 Ix marries4 Peggy5]] or6

[x wants3 [x marries4 Sue7]]],

or to the highest sentential level, yielding

American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982 41

Lenhar t K. Schuber t and Francis Je f f ry Pel le t ier From English to Logic

[(allq x:[x man2]) [x wants3 [x marries4 Peggy5]] or6

(alll x:[x man2]) [x wants3 [x marries4 Sue7]]].

The three readings are logically distinct and all are quite

plausible (in the absence of additional context). The reader

can verify that the first and second readings, but not the

third, could have been obtained by extracting the coordinator

first and the quantifier second.

Finally, we should remark that the distributive rules are not appropr ia te for the group reading of coordi- nate structures in sentences such as

John and Mary carried the sofa ( together) .

We envisage a mereological in te rpre ta t ion in which John and Mary toge ther comprise a t w o - c o m p o n e n t entity. However , we refrain f rom introducing a logical syntax for such entities here (but see the t rea tment of plurals in Schubert , 1982).

5. Parsing

Phrase s t ructure g rammars are relat ively easy to parse. The mos t advanced parser for Gazdar - s ty le grammars that we are aware of is Thompson ' s chart- parser ( T h o m p s o n 1981), which provides for slash categories and coordination, but does not (as of this writing) generate logical translations. We have imple- men ted two small parse r - t rans la tors for prel iminary exper imenta t ion , one wri t ten in S N O B O L and the other in MACLISP. The former uses a recursive de- scent algorithm and generates intensional logic transla- tions. The latter is a ' lef t corner ' parser that uses our reformulated semantic rules to generate convent ional translations. It begins by finding a sequence of left- most phrase-s t ructure-rule branches that lead f rom the first word upward to the sentence node. (e.g., Mary -~ PN -~ NP -~ S). The remaining branches of the phrase structure rules thus selected form a " f ron t i e r " of expectat ions. Next a similar initial-unit sequence is found to connect the second word of the sentence to the lowest- level (most immediate) expectat ion, and so on. There is provision for the definition and use of systems of features, al though we find that the parser needs to do very little feature checking to stay on the right syntactic track. Nei ther parser at present han- dles slash categories and coordinat ion (although they could be handled inefficiently by resort to closure of the grammar under metarules and rule schemata) . Ex- t ract ion of quant if iers f rom the logical - form transla- tions is at present based on the level of syntactic em- bedding and lef t - to- r ight order alone, and no o ther form of postprocessing is a t tempted . l 5

15 Since submission of this paper for publication, we have become aware of several additional papers on parser- t rans la tors similar to ours. One is by Rosenschein & Shieber (1982), another by Gawron et al. (1982); in conception these are based quite directly on the generalized phrase structure grammar of Gazdar and his collaborators, and use reeursive descent parsers. A related Prolog-based approach is described by McCord (1981, 1982).

It has been gratifyingly easy to write these parser- t ranslators , conf i rming us in the convic t ion that Gazda r - s ty l e g rammars hold great promise for the design of natural language unders tanding systems. It is particularly no tewor thy that we found the design of the t rans la tor c o m p o n e n t an a lmost trivial task; no modif icat ion of this componen t will be required even when the parser is expanded to handle slash categories and coordinat ion directly. Encouraged by these re- suits, we have begun to build a full-scale lef t -corner parser. A morphological analyzer that can work with arbi trary sets of formal affix rules is partially imple- mented; this work, as well as some ideas on the con- vent ional t rans la t ion of negat ive adject ive prefixes, plurals, and t e n s e / a s p e c t s t ructure , is r epor ted in Schubert (1982).

6. Concluding Remarks

From the point of view of theoretical and com- puta t ional linguistics, G a z d a r ' s approach to g r a m m a r offers p ro found advan tages over t radi t ional ap- proaches: it dispenses with t r ans fo rmat ions wi thout loss of insight, of fers large linguistic coverage, and couples simple, semant ica l ly wel l -mot iva ted rules of translat ion to the syntact ic rules.

We have a t t empted to show that the advantages of Gazda r ' s approach to g rammar can be secured without c o m m i t m e n t to an intensional ta rget logic for the translat ions of natural language sentences. To moti- vate this endeavour , we have argued that there are phi losophical and pract ical reasons for prefer r ing a convent ional target logic, and that there are as yet no compell ing reasons for abandoning such logics in fav- our of intensional ones. More concre te ly , we have shown how to reformula te Gazda r ' s semantic rules to yield convent ional translations, and have briefly described some extant PSG parsers, including one that is capable of parsing and t ranslat ing in accordance with the reformula ted Gazda r g rammar (minus metal- inguistic constructs) .

We bel ieve that a pa r se r - in t e rp re t e r of this type will prove very useful as the first stage of a natural language unders tanding system. Since the g r ammar rules are expressed in a concise, individually compre- hensible form, such a system will be easy to expand indefinitely. The assignment of a well-defined logical form to input sentences, compat ib le with favoured knowledge representa t ion formalisms, should help to

42 Amer ican Journa l of Computational Linguist ics, Vo lume 8, Number 1, January -March 1982

Lenhart K. Schubert and Francis Jeffry Pelletier From English to Logic

b r i n g a m e a s u r e o f p r e c i s i o n a n d c l a r i t y t o t h e r a t h e r

m u r k y a r e a o f n a t u r a l l a n g u a g e i n t e r p r e t a t i o n b y m a -

c h i n e .

Acknowledgements

T h e a u t h o r s a r e i n d e b t e d t o I v a n Sag f o r a s e r i e s o f

v e r y s t i m u l a t i n g s e m i n a r s h e l d b y h i m a t t h e U n i v e r s i -

ty o f A l b e r t a o n h is l i n g u i s t i c r e s e a r c h , a n d v a l u a b l e

f o l l o w - u p d i s c u s s i o n s . T h e h e l p f u l c o m m e n t s o f t h e

r e f e r e e s a n d o f L o t f i Z a d e h a r e a l s o a p p r e c i a t e d . T h e

r e s e a r c h w a s s u p p o r t e d in p a r t b y N S E R C O p e r a t i n g

G r a n t s A 8 8 1 8 a n d A 2 2 5 2 ; p r e l i m i n a r y w o r k o n t h e

l e f t - c o r n e r p a r s e r w a s c a r r i e d o u t b y o n e o f t h e a u -

t h o r s ( L K S ) u n d e r a n A l e x a n d e r v o n H u m b o l d t f e l -

l o w s h i p in 1 9 7 8 - 7 9 .

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Lenhar t K. Schubert is an associate professor o f

computer science at the University o f Alberta, Edmonton.

He received the Ph.D. degree in computer science f r o m

the University o f Toronto.

Francis Je f f ry Pelletier is a professor o f philosophy at

the University o f Alberta. He received the Ph.D. degree

f r o m the University o f Cali fornia at Los Angeles.

44 American Journal of Computational Linguistics, Volume 8, Number 1, January-March 1982